Sessions keep running across navigation: turn liveness + write-ahead persist

The bug: answering a quiz and navigating away mid-turn, then returning,
re-prompted the same question. Root causes: (1) the answer was durable only
at onFinish — every mid-turn view read the stale pre-answer record;
(2) the server had no notion of a turn in flight; (3) remount treated the
stale record as current, with no re-attach path. Worse, the stale dock let
the quiz be answered again, racing two turns into a last-writer-wins
saveMessages clobber.

Fix (four pieces):
- Write-ahead persist in chat-handler: the request body IS the client's
  canonical state — saved to disk BEFORE the stream starts. The answer is
  durable the instant the request arrives; crashes recover via the existing
  mount continuation.
- Turn liveness registry (src/server/turn-liveness.ts): in-memory chatId →
  in-flight map with a TTL leak guard; beginTurn rejects concurrent turns
  (409) so one turn ever owns a chat. endTurn runs in onFinish AFTER
  persistTurn, so pollers that see not-in-flight always read a finished
  record; a turn-start failure clears the registry (no lockout).
- GET /api/chats/[id] reports turnInFlight; the chat page passes
  initiallyInFlight to the client.
- Client re-attach (components/chat.tsx): while in flight, busy shimmer +
  composer gated + mount continuation suppressed; poll /api/chats/[id] until
  the turn finishes, then syncFromServer adopts the completed transcript.
  Stop stays local-stream only (a server turn has no client request to
  abort).

Verified by scripted-model tests: concurrent turn 409 + registry lifecycle,
write-ahead durability mid-turn, failed-turn lockout guard.

src/server/index.ts

@@ -9,3 +9,4 @@ export * from "./chats.ts";
 export * from "./graph.ts";
 export * from "./jobs.ts";
 export * from "./stores.ts";
+export * from "./turn-liveness.ts";